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HISTORY OF MATHEMATICS • VOLUME 33 Logic’s Lost Genius The Life of Gerhard Gentzen Eckart Menzler-Trott American Mathematical Society • London Mathematical Society Logic’s Lost Genius The Life of Gerhard Gentzen https://doi.org/10.1090/hmath/033 HISTORY OF MATHEMATICS VOLUME 33 Logic’s Lost Genius The Life of Gerhard Gentzen Eckart Menzler-Trott Translated by Craig Smorynski ´ and Edward Griffor Editorial Board American Mathematical Society London Mathematical Society Joseph W. Dauben Alex D. D. Craik Peter Duren Jeremy J. Gray Karen Parshall, Chair Robin Wilson, Chair MichaelI.Rosen This work was originally published in German by Birkh¨auser Verlag under the title Gentzens Problem c 2001. The present translation was created under license for the American Mathematical Society and is published by permission. Photo of Gentzen in hat courtesy of the Archives of the Mathematisches Forschungsinstitut Oberwolfach. Reprinted with permission. 2010 Mathematics Subject Classification. Primary 01A60. For additional information and updates on this book, visit www.ams.org/bookpages/hmath-33 Library of Congress Cataloging-in-Publication Data Menzler-Trott, Eckart. [Gentzens Problem. English] Logic’s lost genius : the life of Gerhard Gentzen / Eckart Menzler-Trott ; translated by Craig Smory´nski and Edward Griffor. p. cm. (History of mathematics ; v. 33) ISBN 978-0-8218-3550-0 (alk. paper) 1. Gentzen, Gerhard. 2. Mathematicians—Germany—Biography. 3. Logic, Symbolic and mathematical. I. Title. QA29 .G467M46 2007 510.92—dc22 2007060550 AMS softcover ISBN: 978-1-4704-2812-9 Copying and reprinting. Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make fair use of the material, such as to copy select pages for use in teaching or research. Permission is granted to quote brief passages from this publication in reviews, provided the customary acknowledgment of the source is given. Republication, systematic copying, or multiple reproduction of any material in this publication is permitted only under license from the American Mathematical Society. Permissions to reuse portions of AMS publication content are handled by Copyright Clearance Center’s RightsLink service. For more information, please visit: http://www.ams.org/rightslink. Send requests for translation rights and licensed reprints to [email protected]. Excluded from these provisions is material for which the author holds copyright. In such cases, requests for permission to reuse or reprint material should be addressed directly to the author(s). Copyright ownership is indicated on the copyright page, or on the lower right-hand corner of the first page of each article within proceedings volumes. c 2007 by the American Mathematical Society. All rights reserved. Reprinted by the American Mathematical Society, 2016. Printed in the United States of America. The American Mathematical Society retains all rights except those granted to the United States Government. ∞ The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. The London Mathematical Society is incorporated under Royal Charter and is registered with the Charity Commissioners. Visit the AMS home page at http://www.ams.org/ 10987654321 212019181716 As a24th probleminmyParis talk I wantedto pose the problem:criteria forthesimplicity ofproof,or, toshowthatcertain proofs aresimplerthan any others. In general, to developatheory ofproofmethods in mathemat- ics. Underagiven set of conditions therecan be but onesimplestproof. Quitegenerally, if there are two proofs foratheorem, youmustkeep going until youhave derived eachfromtheother,oruntil it becomes quiteevi- dent what variant conditions(and aids) have beenusedinthe two proofs. Giventwo routes, it is notright to takeeither ofthese twoortolookfora third; it is necessary to investigate the area lying betweenthe two routes... —DavidHilbert (Hilbert’s notebook, Cod. Ms. D.Hil=600.3) Itis reallyastonishing howmanyhighly gifted logicians diedyoung(Nicod, Herbrand, Gentzen, Spector, Ramsey,etc.). —Kurt G¨odel to Paul Bernays Next I would like tocall your attentionto afrequentlyneglectedpoint, namelythe factthatHilbert’s scheme forthe foundationsofmathematics remains highlyinteresting and important in spiteofmynegative results. What has beenprovedisonlythe specific epistemological objective which Hilbert had in mind cannotbeobtained. Thisobjective was to prove theconsistency ofthe axiomsof classical mathematics onthe basisof ev- idence justasconcrete and immediately convincing aselementary arith- metic. However, viewing thesituationfrom a purely mathematical point ofview,consistencyproofsonthe basisof suitably chosen strongermeta- mathematical presuppositions(as have been givenbyGentzenandothers) arejustas interesting, and they lead to highly important insights into the prooftheoreticstructureofmathematics. —Kurt G¨odel to Constance Reid Contents Preface to the English Edition xiii Introduction xvii 1. Gentzen’s Accomplishments xvii 2.Aimsof My Life Storyof GerhardGentzen xviii 3. Mathematical Logic and National Socialism: The Political Field xix Chapter 1. Early Youth and Abitur 1 1. GerhardGentzen’s Birth 1 2.Gentzen’s Mother:Melanie Gentzen(1873-1968)1 3. Gentzen’s Paternal Grandparents and His Father:The Attorneyand CourtOfficial Dr.jur. Hans Gentzen(1870-1919) 2 4. Youth and Student Days of Hans Gentzen3 5. Gentzen’s Maternal Grandparents: The Physician Alfons Bilharzand Adele Bilharz 4 6. The Shining Example in Gentzen’s Family Tree:The Great, Successful Physician and Natural Scientist Maximilian Theodor Bilharz (1823-1862)9 7.Gentzen’s Sister: Waltraut Sophie MargaretGentzen (*1911) 10 8.The Schoolchild GerhardGentzen11 9. The Death of the Father Means a Move and a NewSchool 12 10. The Beginning of Gentzen’s Intellectual Activity 12 11. Gentzen’s Success at School 15 12.The Abitur on 29February1928 16 Chapter 2.1928-1938—Weimar Republic and National Socialism in Peace. From the Beginning of Studiestothe Extension of the Unscheduled Assistantship for Another Year in Effect from 1October 193821 1. Beginning of StudiesinGreifswald 21 2. Continuation of Study in G¨ottingen 22 3. Is G¨ottingenthe Centre of Mathematics for Gentzen? 23 4. Continuing Studies in Munich 23 5. A Semester in Berlin: Winter Semester 1930/31 26 6. BackinG¨ottingen: SaundersMac Lane and Gentzenasthe “Type of aScientifically OrientedMan”(RichardCourant) 27 7.The Decision: Gentzen’s First Publication, “Ub¨ er die Existenz unabh¨angiger Axiomensysteme zu unendlichenSatzsystemen” and His Programmefor 1932 30 vii viii CONTENTS 8.WhatDoGentzen’s Intellectual Interests and Attitude in 1931 and 1932 Appear to Be?33 9. Political Language in Mathematics in 1918 34 10. Political Language by Hermann Weyl and David Hilbert34 11. Whom Did GentzenKnowinG¨ottingenandWhatDidHe Read? 37 12.Gentzen’s Life in the Early Nazi Period. The Withdrawn Manuscript “Ub¨ er das Verh¨altnis zwischen intuitionistischer und klassischer Arithmetik” of 15 March 1933 38 13. GerhardGentzen’s Dissertation “Untersuchungen¨uber das logische Schließen” of 12 July 1933 41 14.The Penetration of the Nazis into Mathematical Researchatthe University in G¨ottingen 1933 and 1934—Or, VahlenandBieberbach vs. Weber and Wegner 46 15. The State Examination with “Elektronenbahnen in axialsymmetrischen Feldernunter Anwendung auf kosmische Probleme”on16November 1933 51 16. Why Did GentzenJointhe SA? 52 17.Gentzen’s Political Position: A Conjecture 53 18.Gentzen in Financial Difficulties54 19. Financial Straits and Job Hunting 55 20. ConsistencyProoffor Number TheoryinDiscussion with Paul Bernays 57 21. “Widerspruchsfreiheit der reinenZahlentheorie”Mirroredinthe Correspondence of Bernays and Weyl 58 22.Difficultieswiththe “Widerspruchsfreiheit der reinenZahlentheorie” of 11 August 1935 59 23. The Unscheduled Assistantship under Hilbert from 1 November 1935: AProductive Period for Foundational ResearchBegins 62 24. Consistencyof Type Theory63 25. Revising the Proof of the “Widerspruchsfreiheit der reinen Zahlentheorie”64 26. “Die Widerspruchsfreiheit der reinenZahlentheorie”65 27.Gentzen Was an Intellectual Independent 69 28.GentzenExpresses His Thanks to Turing 71 29. Correspondence betweenBernays and Ackermann 1936 to 1940 72 30. The Correspondence betweenBernays and GentzenMerrily Continues 75 31. Invitation to the Parisian DescartesCongress in August 1937.The Invitation to Lecture to the DMV Conference in Bad Kreuznachon 21September 1937:“Die gegenw¨artige Lage in der mathematischen Grundlagenforschung”. The Extension of the Tenure of the Unscheduled Assistantship on 1 October 1937for aYear77 32.Jean Cavaill`esandGerhardGentzen 82 33. GentzenBecomes an “Associate”of the Publication of Scholz’s Forschungen zur Logik und zur Grundlegung der exakten Wissenschaften 84 34.“Die gegenw¨artige Lage in der mathematischenGrundlagenforschung” 86 35. “Neue Fassung desWiderspruchsfreiheitsbeweisesder reinen Zahlentheorie” 193888 CONTENTS ix 36. Bernays’ Viewof Gentzen’s Programme (Second ConsistencyProof) 89 37.Correspondence with Paul Bernays 93 38.Extension of the Unscheduled Position for Another Year Taking Effect1October 1938 97 39. Bernays’ Views of Hilbert’s Programme and Gentzen’s Place in It 100 40. Closing Thoughts on Paul Bernays 101 41. Longer Notes102 Chapter 3. 1939-1942—From the Beginning of the War to Dismissal from the Wehrmacht and the Wartime Habilitation under Helmut Hasse 117 1. 1939: At the Highpoint of Reputation 117 2.The Second Volume of Grundlagen der
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    Quadratic Forms and Their Applications Proceedings of the Conference on Quadratic Forms and Their Applications July 5{9, 1999 University College Dublin Eva Bayer-Fluckiger David Lewis Andrew Ranicki Editors Published as Contemporary Mathematics 272, A.M.S. (2000) vii Contents Preface ix Conference lectures x Conference participants xii Conference photo xiv Galois cohomology of the classical groups Eva Bayer-Fluckiger 1 Symplectic lattices Anne-Marie Berge¶ 9 Universal quadratic forms and the ¯fteen theorem J.H. Conway 23 On the Conway-Schneeberger ¯fteen theorem Manjul Bhargava 27 On trace forms and the Burnside ring Martin Epkenhans 39 Equivariant Brauer groups A. FrohlichÄ and C.T.C. Wall 57 Isotropy of quadratic forms and ¯eld invariants Detlev W. Hoffmann 73 Quadratic forms with absolutely maximal splitting Oleg Izhboldin and Alexander Vishik 103 2-regularity and reversibility of quadratic mappings Alexey F. Izmailov 127 Quadratic forms in knot theory C. Kearton 135 Biography of Ernst Witt (1911{1991) Ina Kersten 155 viii Generic splitting towers and generic splitting preparation of quadratic forms Manfred Knebusch and Ulf Rehmann 173 Local densities of hermitian forms Maurice Mischler 201 Notes towards a constructive proof of Hilbert's theorem on ternary quartics Victoria Powers and Bruce Reznick 209 On the history of the algebraic theory of quadratic forms Winfried Scharlau 229 Local fundamental classes derived from higher K-groups: III Victor P. Snaith 261 Hilbert's theorem on positive ternary quartics Richard G. Swan 287 Quadratic forms and normal surface singularities C.T.C. Wall 293 ix Preface These are the proceedings of the conference on \Quadratic Forms And Their Applications" which was held at University College Dublin from 5th to 9th July, 1999.
  • Gödel on Finitism, Constructivity and Hilbert's Program

    Gödel on Finitism, Constructivity and Hilbert's Program

    Lieber Herr Bernays!, Lieber Herr Gödel! Gödel on finitism, constructivity and Hilbert’s program Solomon Feferman 1. Gödel, Bernays, and Hilbert. The correspondence between Paul Bernays and Kurt Gödel is one of the most extensive in the two volumes of Gödel’s collected works devoted to his letters of (primarily) scientific, philosophical and historical interest. It ranges from 1930 to 1975 and deals with a rich body of logical and philosophical issues, including the incompleteness theorems, finitism, constructivity, set theory, the philosophy of mathematics, and post- Kantian philosophy, and contains Gödel’s thoughts on many topics that are not expressed elsewhere. In addition, it testifies to their life-long warm personal relationship. I have given a detailed synopsis of the Bernays Gödel correspondence, with explanatory background, in my introductory note to it in Vol. IV of Gödel’s Collected Works, pp. 41- 79.1 My purpose here is to focus on only one group of interrelated topics from these exchanges, namely the light that ittogether with assorted published and unpublished articles and lectures by Gödelthrows on his perennial preoccupations with the limits of finitism, its relations to constructivity, and the significance of his incompleteness theorems for Hilbert’s program.2 In that connection, this piece has an important subtext, namely the shadow of Hilbert that loomed over Gödel from the beginning to the end of his career. 1 The five volumes of Gödel’s Collected Works (1986-2003) are referred to below, respectively, as CW I, II, III, IV and V. CW I consists of the publications 1929-1936, CW II of the publications 1938-1974, CW III of unpublished essays and letters, CW IV of correspondence A-G, and CW V of correspondence H-Z.